Consequences of Isoperimetry
A simple integration over r (or sum in the case of graphs) shows that a d-dimensional isoperimetric inequality implies a d-dimensional volume growth, namely
where B(x,r) denotes the ball of radius r around the point x in the Riemannian distance or in the graph distance. In general, the opposite is not true, i.e. even uniformly exponential volume growth does not imply any kind of isoperimetric inequality. A simple example can be had by taking the graph Z (i.e. all the integers with edges between n and n + 1) and connecting to the vertex n a complete binary tree of height |n|. Both properties (exponential growth and 0 isoperimetric dimension) are easy to verify.
An interesting exception is the case of groups. It turns out that a group with polynomial growth of order d has isoperimetric dimension d. This holds both for the case of Lie groups and for the Cayley graph of a finitely generated group.
A theorem of Varopoulos connects the isoperimetric dimension of a graph to the rate of escape of random walk on the graph. The result states
Varopoulos' theorem: If G is a graph satisfying a d-dimensional isoperimetric inequality then
where is the probability that a random walk on G starting from x will be in y after n steps, and C is some constant.
Read more about this topic: Isoperimetric Dimension
Famous quotes containing the words consequences of and/or consequences:
“There is not much that even the most socially responsible scientists can do as individuals, or even as a group, about the social consequences of their activities.”
—Eric J. Hobsbawm (b. 1917)
“Cultivate the habit of thinking ahead, and of anticipating the necessary and immediate consequences of all your actions.... Likewise in your pleasures, ask yourself what such and such an amusement leads to, as it is essential to have an objective in everything you do. Any pastime that contributes nothing to bodily strength or to mental alertness is a totally ridiculous, not to say, idiotic, pleasure.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)